Prefix grammars: an alternative characterization of the regular languages

We consider a class of length-preserving string rewriting systems and show that the set of encodings of pairs of strings 〈s,f〉 such that f can be derived from s using the rewriting rules can be accepted by finite automata. As a consequence, we show the existence of a linear time algorithm for determining the solvability of a given k×n peg-solitaire board, for any fixed k. This result is in contrast to the results of (13) and (1) that the same problem is NP-hard for n×n boards. We look at some related string rewriting systems and find conditions under which the encodings of the pairs 〈s,f〉 where f can be derived from s is regular.

We propose a new extension of splicing systems, called multiple splicing systems, based on a kind of logic grammars. First, we introduce a class of Elementary formal systems, called simple H-form EFS, and show that its generative power is equivalent to the class of basic splicing systems (the original Head's splicing system) and is more adequate as a representation device for formal languages. Next, we gradually extend the simple class of EFSs and get a very natural extension of splicing systems, multiple splicing systems. We show that multiple splicing systems have universal computability.

A notion of derivation tree is introduced for ground term rewriting systems. Using standard tree language theory, new proofs are given for some old results.